Average Error: 58.1 → 0.7
Time: 14.4s
Precision: 64
\[\frac{e^{x} - e^{-x}}{2}\]
↓
\[\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}\]
\frac{e^{x} - e^{-x}}{2}
↓
\frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}
double f(double x) {
        double r1491714 = x;
        double r1491715 = exp(r1491714);
        double r1491716 = -r1491714;
        double r1491717 = exp(r1491716);
        double r1491718 = r1491715 - r1491717;
        double r1491719 = 2.0;
        double r1491720 = r1491718 / r1491719;
        return r1491720;
}
↓
double f(double x) {
        double r1491721 = x;
        double r1491722 = 5.0;
        double r1491723 = pow(r1491721, r1491722);
        double r1491724 = 0.016666666666666666;
        double r1491725 = 0.3333333333333333;
        double r1491726 = r1491725 * r1491721;
        double r1491727 = 2.0;
        double r1491728 = fma(r1491726, r1491721, r1491727);
        double r1491729 = r1491728 * r1491721;
        double r1491730 = fma(r1491723, r1491724, r1491729);
        double r1491731 = r1491730 / r1491727;
        return r1491731;
}

Error

Bits error versus x

Derivation

  1. Initial program 58.1

    \[\frac{e^{x} - e^{-x}}{2}\]

  2. Taylor expanded around 0 0.7

    \[\leadsto \frac{\color{blue}{2 \cdot x + \left(\frac{1}{3} \cdot {x}^{3} + \frac{1}{60} \cdot {x}^{5}\right)}}{2}\]

  3. Simplified0.7

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}}{2}\]

  4. Final simplification0.7

    \[\leadsto \frac{\mathsf{fma}\left({x}^{5}, \frac{1}{60}, \mathsf{fma}\left(\frac{1}{3} \cdot x, x, 2\right) \cdot x\right)}{2}\]

Reproduce

herbie shell --seed 2019144 +o rules:numerics
(FPCore (x)
  :name "Hyperbolic sine"
  (/ (- (exp x) (exp (- x))) 2))